Primitive of $\frac{\ln (x^2 + 1)}{2x^2 \ln (x + 1)}$

Question

I want to integrate $ \displaystyle \int \dfrac{\ln (x^2 + 1)}{2x^2 \ln (x + 1)} \ dx $

I've tried a $x \mapsto \frac{1}{x} $ substitution, but that complicates the integrand.

Mathematica's online integrator could not integrate it, however I have heard that sometimes it is possible to evaluate integrals that Mathematica cannot.

Comment on the edit:

In light of the comment on the definite integral's convergence, I have modified the question to an indefinite integral. Apologies for the inconvenience.